Check predictions with residuals · ACT Math

A fitted model supplies predictions, not guaranteed observations. A residual is observed minus predicted and can be positive or negative.

Choose a representation

Evaluate the model at the actual input before subtracting. Compare fits on the same observations using a stated loss, such as the sum of squared residuals.

Worked example: follow the quantity

A fitted model predicts y=3 x+5. At x=3, observed y is 15. What is the residual (observed minus predicted)? Evaluate the model using the stated input. Subtract that prediction from the actual observed output. Substitute the given numbers: 15-(3 × 3+5) = 1. Therefore the requested value is 1.

Contrast a plausible mistake

For that example, the tempting answer -1 follows this error: Subtract observed from predicted, reversing the residual sign. The requested quantity is 1. Explain why the wrong operation changes the mathematical object, not merely the arithmetic.

A second representation

For x=0,1,2,3 a measured output is 5,6,9,14. Model L(x)=5+3 x and model Q(x)=5+x² are compared at x=2. By how much is L’s absolute error greater than Q’s? Use the observed output at input 2 for both models. Compare absolute errors, not model values alone. Substitute the given numbers: ((5+3 × 2)-(5+4))-0 = 2. Therefore the requested value is 2.

Transfer before checking

A fitted model predicts y=3 x+6. At x=4, observed y is 20. What is the residual (observed minus predicted)? Try without viewing the answer. Check: 2. Evaluate the model using the stated input. Subtract that prediction from the actual observed output. Substitute the given numbers: 20-(3 × 4+6) = 2.

Scope and conditions

A negative residual means the point is below the prediction. Squaring removes its sign but amplifies large misses. Correlation and a small residual do not establish causation. These original practice tasks sample the official topic; they do not certify complete coverage or a predicted ACT score.

WORKED EXAMPLE

The diagram shows one observation and the fitted line. What is the residual, defined as observed minus predicted, at the marked point?

Fitted line y=3x+5 and observed point (3,15). The line predicts 14 at x=3.

Evaluate the model using the stated input.
Subtract that prediction from the actual observed output.
Substitute the given numbers: 15-(3 × 3+5) = 1.
Therefore the requested value is 1.

Answer: 1

Compare the other answer choices

Subtract only the slope term and forget the model intercept.

Report the predicted output instead of the residual.

-1

Subtract observed from predicted, reversing the residual sign.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

ACT Mathematics content